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Theorem f1orn 6616
Description: A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)
Assertion
Ref Expression
f1orn (𝐹:𝐴1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹))

Proof of Theorem f1orn
StepHypRef Expression
1 dff1o2 6611 . 2 (𝐹:𝐴1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = ran 𝐹))
2 eqid 2824 . . 3 ran 𝐹 = ran 𝐹
3 df-3an 1086 . . 3 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ ((𝐹 Fn 𝐴 ∧ Fun 𝐹) ∧ ran 𝐹 = ran 𝐹))
42, 3mpbiran2 709 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹))
51, 4bitri 278 1 (𝐹:𝐴1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1084   = wceq 1538  ccnv 5541  ran crn 5543  Fun wfun 6337   Fn wfn 6338  1-1-ontowf1o 6342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-in 3926  df-ss 3936  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350
This theorem is referenced by:  f1f1orn  6617  infdifsn  9117  efopnlem2  25251  cycpmcl  30790
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